Philosophy assignment 10 questions

“If the inference from p to q is valid, and the inference from q to r is valid, then the inference from p to r must be valid as well.”

I agree!

I disagree!

PHIL 110; Spring 2019; Lecture 20 1

“If two objects are indistinguishable, then it can’t be true that one of them is red and also true that the other is not red.”

I agree!

I disagree!

PHIL 110; Spring 2019; Lecture 20 2

1 4 : B i v a l e n c e P H I L 1 1 0 ; S p r i n g 2 0 2 0 ; To m D o n a l d s o n

Bivalence

• In this course, we’ve been assuming that every statement is either true or false.

• To put it another way, we’ve been assuming that given any statement, either it or its negation is true.

• This is called the “principle of bivalence” (or even the “law of bivalence”).

• In this lecture, we’ll look at some objections to the principle of bivalence, and discuss how to cope.

PHIL 110; Spring 2020; Lecture 14 4

1 : Va g u e Te r m s

Vagueness

• Suppose we have a sequence of 1000 tiles. We can call them “Tile 1”, “Tile 2”, “Tile 3”, … , “Tile 1000”.

• Tile 1 is the colour of the leaf on the Canadian flag.

• Each tile in the sequence is a little bit less red than its predecessor – but the differences are imperceptibly small. Adjacent tiles in the sequence are indistinguishable.

• Tile 1000 is the colour of a pumpkin.

PHIL 110; Spring 2020; Lecture 14 6

Vagueness • The principle of bivalence tells us that every tile in the sequence is

either truly describable as “red” or truly describable as “not red” – like this:

• But, arguably, this is not plausible: • There are tiles in the middle which we wouldn’t call “red” but which we also

wouldn’t call “not red”. • It isn’t credible that there are adjacent tiles, one of which is truly describable

as “red” and one of which is truly describable as “not red”.

PHIL 110; Spring 2020; Lecture 14 7

Vagueness

• Here, arguably, is a more attractive account. The tiles at the beginning of the sequence are properly called “red”. The tiles at the end of the sequence are properly called “not red”. Then there are some tiles in the middle which have a third, intermediate status. These tiles can’t truly be described as “red”, but they can’t truly be described as “not red” either:

PHIL 110; Spring 2020; Lecture 14 8

Some more vague terms

• To use the philosophical jargon, “red” is vague.

• Here are some other vague terms: • “grownup”

• “cold day”

• “too much ice cream to eat in one sitting”

PHIL 110; Spring 2020; Lecture 14 9

2 : S t a t e m e n t s A b o u t t h e F u t u r e

The Correspondence Theory of Truth

• Some philosophers think that a statement is true just in case it correctly depicts a fact – some chunk of reality.

PHIL 110; Spring 2020; Lecture 14 11

Oscar is in the guitar case.

PHIL 110; Spring 2020; Lecture 14 12

Oscar is next to the flowers.

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The Correspondence Theory of Truth

• Some philosophers think that a statement is true just in case it correctly depicts a fact – some chunk of reality.

• A sentence is false, on this view, if its negation correctly depicts a fact.

PHIL 110; Spring 2020; Lecture 14 14

Oscar is in a red cupboard.

PHIL 110; Spring 2020; Lecture 14 15

Statements About the Future

• Suppose we accept the correspondence theory of truth.

• Suppose we also accept the claim that the future doesn’t yet exist. • Now consider the statement “Canada will win an odd number of medals in the

2020 Olympic Games.”

• Arguably, this statement isn’t true now (because there is currently no fact to which it corresponds).

• And arguably, this statement isn’t false now (because there is currently no fact to which its negation corresponds).

• If this is right, then the statement is a counterexample to the principle of bivalence. Our statement has some third status, “open,” perhaps, or “unsettled.”

• Many have attributed this view to Aristotle – although the attribution is contentious.

PHIL 110; Spring 2020; Lecture 14 16

3 : T h e L i a r P a r a d o x

The Liar Paradox

Consider the following sentence:

The red sentence on slide 18 is false.

• If we say that this sentence is true, this implies that the sentence is false, which is a contradiction!

• If we say that the sentence is false, we’re saying that it’s false that the sentence is false – i.e. that the sentence is true. This is a contradiction again!

• Perhaps it’s best to refrain from saying that the sentence is either true nor false!

PHIL 110; Spring 2019; Lecture 20 18

4 : R e f o r m i n g S t a t e m e n t L o g i c

Bivalence

• As I said, the classical approach to logic assumes the principle of bivalence – we assume that every statement is either true or false.

• However, there are apparent counterexamples to this thesis: • Statements involving vague words.

• Statements about the future.

• Paradoxes

• Others?

• Perhaps we need to reform logic in order to accommodate such cases …

PHIL 110; Spring 2020; Lecture 14 20

Three-Valued Logic

• Suppose we accept the view that there are really three truth values, not two. Some statements are true; some are false; some are intermediate.

• Our truth tables will get bigger! For each binary connective, we now need a truth table with nine rows instead of just four.

• How will we fill in the rows?

• This is contentious – I will present one approach.

PHIL 110; Spring 2020; Lecture 14 21

Conjunction

• A conjunction is true when both conjuncts are true.

• A conjunction is false when either one of its conjuncts is false.

• Otherwise, the conjunction is intermediate.

PHIL 110; Spring 2020; Lecture 14 22

Conjunction

PHIL 110; Spring 2020; Lecture 14 23

p q (p & q)

T T T

T I I

T F F

I T I

I I I

I F F

F T F

F I F

F F F

Disjunction

• A disjunction is true when either one of the disjuncts is true.

• A disjunction is false when both of the disjuncts are false.

• Otherwise, the disjunction is intermediate.

PHIL 110; Spring 2020; Lecture 14 24

Disjunction

PHIL 110; Spring 2020; Lecture 14 25

p q (p  q)

T T T

T I T

T F T

I T T

I I I

I F I

F T T

F I I

F F F

Negation

• If a statement is true, its negation is false.

• If a statement is false, its negation is true.

• If a statement is intermediate, its negation is also intermediate.

PHIL 110; Spring 2020; Lecture 14 26

p p

T F

I I

F T

Conditionals

• (p → q) is equivalent to (p  q).

• (p ↔ q) is equivalent to ((p → q) & (q → p)).

PHIL 110; Spring 2020; Lecture 14 27

A New Connective

• We’ve seen that in our new logic we have to reform the truth tables for the familiar connectives.

• We can also introduce some new connectives! For example, we could introduce a connective 𝕀 meaning “It is neither true nor false that …” with the following truth table:

PHIL 110; Spring 2019; Lecture 20 28

p 𝕀p

T F

I T

F F

The Definition of Validity

• If we assume the principle of bivalence, we will regard these definitions as equivalent: • An inference is valid just in case there is no possible situation in which the

premises are true and the conclusion false.

• An inference is valid just in case there is no possible situation in which the premises are true and the conclusion is not true.

• But now we must regard these definitions as non-equivalent…

• Which should we choose?

• Well, let’s see what happens if we choose the first definition …

PHIL 110; Spring 2020; Lecture 14 29

The Definition of Validity Consider the following three statements:

A 𝕀A (A & 𝕀A) T F T I T F F F F

Given our definition of validity, we have to say that from 𝕀A you can validly infer A, and from A you can validly infer (A & 𝕀A) … but you can’t validly infer (A & 𝕀A) from 𝕀A!!

This is absurd – so we have to reject this definition of validity.

PHIL 110; Spring 2020; Lecture 14 30

The Definition of Validity

• We have to choose between two definitions of validity: • An inference is valid just in case there is no possible situation in which the

premises are true and the conclusion false.

• An inference is valid just in case there is no possible situation in which the premises are true and the conclusion is not true.

• The first definition turns out to be ridiculous.

• So we have to choose the second.

PHIL 110; Spring 2020; Lecture 14 31

Which of our rules are valid?

• It’s easy to check that many of our natural deduction rules are still valid in the new system. For example, Conj and Simp are still valid!

• However, some of our natural deduction rules have to be rejected – or at least, reformed.

• Consider, for example, CP.

PHIL 110; Spring 2019; Lecture 20 32

Conditional Proof • Consider the following proof in our natural deduction system:

1. A Prem │ 2. B Supp/CP │ 3. (A & B) 1, 2 Conj

4. (B → (A & B)) 2-3,CP

• The inference from A to (B → (A & B)) is not valid in our new system.

• So (within our new system) we must say that there is something wrong with the above proof.

• But Conj is valid in our new system, as I said.

• So we have to reject CP – or at least reform it somehow.

PHIL 110; Spring 2019; Lecture 20 33

Reductio ad Absurdum • Consider this proof in our natural deduction system:

1. A Prem │2. (B & B) Supp/RA │3. ⊥ 2, R 4. (B & B) 2-3, RA

• The inference from A to (B & B) is not valid in our system.

• So (within our new system) we must reject the above proof.

• So we must reject (or at least reform) the RA rule.

PHIL 110; Spring 2019; Lecture 20 34

A Complaint About the New System

• Arguably, CP and RA are essential to mathematics.

• We can’t live without them.

• Thus, the new logic can’t be accepted.

PHIL 110; Spring 2019; Lecture 20 35

5 : A n U n s o l v e d P r o b l e m

An Unsolved Problem

• We’ve seen that the principle of bivalence is problematic.

• However, our new logic has its own problems!

• Options: • We could defend the principle of bivalence from the objections.

• We could accept that the principle of bivalence is mistaken, and then offer some alternative defence of our natural deduction rules.

• We could learn to live with our three-valued logic.

• We could find an altogether new logic …

• All of these approaches have their defenders!

PHIL 110; Spring 2020; Lecture 14 37

A Problem to Finish

Show that for any statement p one can construct a natural deduction proof of the following statement:

(p  p)

Presumably, if we reject the law of bivalence we will also deny that all instances of (p  p) are true (Do you agree?). So we will wish to reject at least one of the rules in your proof. Which one do you think we should reject?

PHIL 110; Spring 2019; Lecture 20 38